Noncomutative Discrete Geometric Analysis

非计算离散几何分析

• 批准号: 18540221
• 负责人: SUNADA Toshikazu
• 金额: $ 2.57万
• 依托单位: Meiji University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2006
• 资助国家: 日本
• 起止时间: 2006 至 2007
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-18540221/
• 关键词: Discrete Laplacian; Graph; Crystal lattice; Diamond twin; Random walk; Ctrystal lattice; 結晶格子; 格子振動; K_4結晶; 離散的ラブラシアン; 大偏差原理; ランダムウォーク

Discrete geometric analysis is a hybrid field of several traditional disciplines: graph theory, geometry, theory of discrete groups, and probability. As indicated by the title, this field concerns solely analysis on graphs, a synonym of "1-dimensional cell complex". Edges in a graph as 1-dimensional objects, however, do not play a substantial role except when we discuss geometry of graphs. Therefore our view is different from, for instance, the case of quantum graphs where differential operators on edges are vital. Actually the role of edges in analysis is just to give a neighboring relation among vertices, and difference operators linked with this relation replace differential operators. We thus do not need to worry about irregularity of functions which, for differential operators, may cause some trouble, if not serious. Instead, the combinatorial aspect of graphs creates a different kind of technical complication. Furthermore, analysis on both infinite graphs and non-compact manifold … More s involves much the same degree of difficulty.The "protagonist" in our research is a discrete analogue of Laplacians on manifolds (and related operators), which appears in many parts of mathematical sciences. In the nature of things, ideas cultivated in global analysis provide us a good guiding principle on the one hand, and the practical motivation is the moving force of theoretical progress on the other..Discrete Laplacians appear in both geometric crystallography and probability.In our research, we applied a remarkable result in the theory of random walks on crystal lattices to pin down a diamond crystal. It is interesting to rephrase the result such as "A random walker on a crystal lattice may detect the most natural way for his crystal lattice to sit in space". A diamond twin means a hypothetical crystal which shares the property of symmetry satisfied by the diamond crystal. Our result claims that there is only diamond twin, which is obtained as the standard realization of the maximal abelian covering graph of the complete graph K_4.As for the activity, I was a member of organizers of the special project held at Newton Institute, Cambridge University which stated from January and ended up in June, 2007. Less

离散几何分析是几个传统学科的混合领域：图论、几何学、离散群论和概率论。如标题所示，该领域仅涉及对图的分析，图是“一维细胞复合体”的同义词。然而，图中的边作为一维对象并没有起到实质性的作用，除非我们讨论图的几何。因此，我们的观点不同于量子图的情况，在量子图中，边上的微分算符是至关重要的。实际上，边在分析中的作用就是给出顶点之间的一种邻接关系，而与这种关系相联系的差分算子就代替了微分算子。因此，我们不需要担心函数的不规则性，对于微分算子来说，这可能会造成一些麻烦，如果不严重的话。相反，图形的组合方面创造了一种不同的技术复杂性。此外，对无限图和非紧流形…的分析更多的S涉及的困难程度大致相同。我们研究的“主角”是流形(及其相关算子)上的拉普拉斯的离散类比，它出现在数学科学的许多部分。在事物的本质上，整体分析中培育的思想一方面为我们提供了很好的指导原则，另一方面，实践动力是理论进步的推动力。离散拉普拉斯理论在几何结晶学和概率论中都出现了。在我们的研究中，我们将晶体晶格上的随机游动理论应用于钻石晶体的固定。有趣的是，将结果重新表述为“晶格上的随机漫游者可以探测到他的晶格坐在空间中最自然的方式”。钻石孪晶是指具有钻石晶体所满足的对称性的假想晶体。我们的结果表明，只有钻石孪生的存在，这是作为完全图K_4的最大阿贝尔覆盖图的标准实现而得到的。至于这次活动，我是剑桥大学牛顿研究所特别项目的组织者之一，该项目于2007年1月至6月结束。较少

项目成果

Mathematical theory of lattice vibrations

晶格振动的数学理论

• 发表时间: 2006
• 期刊: Pure and Appl. Math. Quaterly 2
• 作者: Shubin;T.Sunada
• 通讯作者: T.Sunada

On the K_4crystal

在K_4水晶上

• 发表时间: 2007
• 作者: M.Kotani;T.Sunada;T.Sunada
• 通讯作者: T.Sunada

Large deviation and the tangent cone at infinity of a crystal lattice

• DOI: 10.1007/s00209-006-0951-9
• 发表时间: 2006-03
• 期刊: Mathematische Zeitschrift
• 影响因子: 0.8
• 作者: M. Kotani;T. Sunada

Crystals that nature might miss creating

大自然可能错过的晶体

• 发表时间: 2008
• 期刊: Notices Amer. Math. Soc. 55
• 作者: Mario Roy;Hiroki Sumi;Mariusz Urbanski;H. Sumi;R. Stankewitz and H. Sumi;H. Sumi;H. Sumi;R. Stankewitz and H. Sumi;H. Sumi;H. Sumi;角 大輝;H. Sumi and M. Urbanski;H. Sumi and M. Urbanski;角大輝;H. Sumi;H. Sumi;H. Sumi;Hiroki Sumi;H. Sumi;角大輝;角大輝;H. Sumi;角大輝;角大輝;H. Sumi;角大輝;角大輝;角大輝;角大輝;H. Sumi;T. Sunada;T. Sunada
• 通讯作者: T. Sunada